If the universe stops expanding, will time run backwards?

In other words, there is a perceived arrow of time, and there is a thermodynamic arrow of time, and they both always point in the forward direction. Is this causation? While some — notably Sean Carroll — speculate that they are linked in some fashion, we should remember that is pure speculation, and that no link has ever been uncovered or demonstrated. As far as we can tell, the thermodynamic arrow of time is a consequence of statistical mechanics, and is a property that emerged for many-body systems. (You might need at least three.) The perceived arrow of time, however, seems largely independent of anything entropy or thermodynamics may do.

What, if anything, happens when we bring the expanding Universe into the equation?

It’s true that, for all of time since (at least) the hot Big Bang, the Universe has been expanding. It’s also true that while time is linear, passing at that constant perceived rate of one second per second, the rate at which the Universe expands is not. The Universe expanded much more quickly in the past, is expanding more slowly today, and will asymptote to a finite, positive value. This, as far as we understand it, means that distant galaxies that aren’t gravitationally bound to us will continue to recede from our perspective, faster and faster, until what remains of our Local Group is the only remaining thing we can access.